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Ostensibly is used to meant that something ought to be true, but either the speaker can't verify its truth, or the speaker suspects it may not be true:

Dr. Jones is, ostensibly, the primary authority on the subject, but nevertheless he often defers to the opinions of other experts.

In the United States the Justice Department is ostensibly independent from the executive, but recent actions by the current president belie its supposed autonomy.

Purportedly is used to indicate something is presented as true, when it is actually false.

Government documents purportedly stated that New Orleans' levies were able to withstand flooding of much greater magnitude than happened after hurricane Katrina in 2005, but the truth was that they were nowhere near enough.

He was said to be the richest man in the country, but, in actuality, his purported wealth existed mostly in his own head.

These are loose synonyms, but purported more strongly indicates deception. For example, in the above example I could change out "supposed" with "purported" to suggest that the government has been actively lying about the Justice Department's independence from the executive branch:

... but recent actions by the current president belie its purported autonomy

Ostensibly is used to meant that something ought to be true, but either the speaker can't verify its truth, or the speaker suspects it may not be true:

Dr. Jones is, ostensibly, the primary authority on the subject, but nevertheless he often defers to the opinions of other experts.

In the United States the Justice Department is ostensibly independent from the executive, but recent actions by the current president belie its supposed autonomy.

Purportedly is used to indicate something is presented as true, when it is actually false.

Government documents purportedly stated that New Orleans' levies were able to withstand flooding of much greater magnitude than happened after hurricane Katrina in 2005, but the truth was that they were nowhere near enough.

He was said to be the richest man in the country, but in actuality his purported wealth existed mostly in his own head.

These are loose synonyms, but purported more strongly indicates deception.

Ostensibly is used to meant that something ought to be true, but either the speaker can't verify its truth, or the speaker suspects it may not be true:

Dr. Jones is, ostensibly, the primary authority on the subject, but nevertheless he often defers to the opinions of other experts.

In the United States the Justice Department is ostensibly independent from the executive, but recent actions by the current president belie its supposed autonomy.

Purportedly is used to indicate something is presented as true, when it is actually false.

Government documents purportedly stated that New Orleans' levies were able to withstand flooding of much greater magnitude than happened after hurricane Katrina in 2005, but the truth was that they were nowhere near enough.

He was said to be the richest man in the country, but, in actuality, his purported wealth existed mostly in his own head.

These are loose synonyms, but purported more strongly indicates deception. For example, in the above example I could change out "supposed" with "purported" to suggest that the government has been actively lying about the Justice Department's independence from the executive branch:

... but recent actions by the current president belie its purported autonomy

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Ostensibly is used to meant that something ought to be true, but either the speaker can't verify its truth, or the speaker suspects it may not be true:

Dr. Jones is, ostensibly, the primary authority on the subject, but nevertheless he often defers to the opinions of other experts.

In the United States the Justice Department is ostensibly independent from the executive, but recent actions by the current president belie its supposed autonomy.

Purportedly is used to indicate something is presented as true, when it is actually false.

Government documents purportedly stated that New Orleans' levies were able to withstand flooding of much greater magnitude than happened after hurricane Katrina in 2005, but the truth was that they were nowhere near enough.

He was said to be the richest man in the country, but in actuality his purported wealth existed mostly in his own head.

These are loose synonyms, but purported more strongly indicates deception.