a new material [that] they say is as soft as cotton

Question

Can the relative pronoun "that" be omitted in the following?

A group of researchers has developed a new material that they say is as soft as cotton but as strong as Kevlar and as conductive as many metals.

Answer 1

A group of researchers has developed a new material that they say is as soft as cotton but as strong as Kevlar and as conductive as many metals.

they say is an interruption and if removed should not affect the grammar of the sentence. For clarity, we remove this interruption and get

A group of researchers has developed a new material that is as soft as cotton but as strong as Kevlar, and as conductive as many metals. [comma added as suggested by @Jack O'Flaherty]

that is the subject of the relative clause and hence cannot be deleted alone.

I read the book that was sent to me last Friday.

Similarly, that is the subject of the relative clause and hence cannot be deleted alone.

There are ways to reduce relative clauses with relative pronouns acting as subjects, as described below.

Restrictive Relative Clauses

Subject pronouns can be deleted if –ing is added to the verb.

I like the paintings that hang in the SASB North lobby.

I like the paintings hanging in the SASB North lobby.

Non-restrictive Relative Clauses

Subject pronouns with “be” verbs can be deleted in non-restrictive clauses.

I am moving to Louisville, KY, which is home to the Muhammad Ali Museum.

I am moving to Louisville, KY, home to the Muhammad Ali Museum.

My mother, who is an excellent cook, is thinking of opening a restaurant.

My mother, an excellent cook, is thinking of opening a restaurant.

More examples are found in this link.

The product, which seemed perfect in many ways, failed to succeed in the market. Reduced: The product, perfect in many ways, failed to succeed in the market.

The boy who was pleased by his grades went out with his friends to celebrate. Reduced: The boy, pleased by his grades, went out with his friends to celebrate.

The box, which was on the table, was made in Italy. Reduced: The box on the table was made in Italy.

The woman who was at the meeting spoke about business in Europe. Reduced: The woman at the meeting spoke about business in Europe.

As shown above, relative clauses with their relative pronouns acting as subjects may be reduced in certain situations, in certain ways. Their relative pronouns cannot be deleted alone.