This day would constitute the greatest challange our production would ever face.
Why is WOULD used in this sentence?
Source: https://youtu.be/ivfspJOAtBI at 12:54.
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Consider the following sentence:
This day will constitute the greatest challenge our production will ever face.
The helper verb "to will" is used to indicate future tense of the other verbs, in this example to constitute and to face. Some other languages may not need a helper verb to change a verb to future tense (for example in French one would write constituera in the place of the English will constitute), but in English it works like this, with a helper verb.
Now consider that something will happen in the future within a story that I am telling you:
"Yesterday I was hungry. I was in the kitchen and I saw some tomatoes and some spaghetti lying on the table. I would cook them and then I would eat them and it would take me 30 minutes to eat the meal. An hour later I was washing the dishes with my belly full of goodness."
I could not simply say
I will cook them because I am telling the story in the past tense and I am telling you about something that lies in the future in the context of my story, but not necessarily in the future when I am telling the story.
You may say, okay why not just tell me what happened chronologically? Why tell me about something that will happen if it has happened already? It adds an interesting quality to the story: it adds uncertainty to the storyline. After all, if I make a statement about the future it is just that: an anticipation or prediction. What actually happens after all may differ from the initial prediction. This is true for any statement I make now about the future, but is also true for a statement about the future within the context of a story. An alternative ending to my story above could be:
... An hour later I gave up cooking anything because my stove didn't work and I was staring at cold water sitting in a pot. Instead of wasting more time I ordered pizza and made myself a tomato salad while waiting for the delivery guy to ring the bell.
The same is true for the example you gave: It is part of a story told in the past tense, telling you about something that lies in the future at that particular time in the story. I would have to watch the video to find out if the prediction was accurate, or if that day turned out as challenging as expected.
In other words, the verb will is conjugated to its past tense would because the storyteller is telling you about something that was going to happen in the story, which is set in the past.
I hope this helps. If anything remains unclear please let me know in the comments!