claim/ claim to

The government is sending a team of agronomists to remote areas to look into cooperative farming techniques that the peasants claim have greatly improved their crop yields.

I think the word "to" is missing in the sentence above.

The government is sending a team of agronomists to remote areas to look into cooperative farming techniques that the peasants claim to have greatly improved their crop yields.

Am I correct? Please advise.

Thanks for the explanation. However, I still would like to say what I think of this sentence. Here is my two cents:

The government is sending a team of agronomists to remote areas to look into cooperative farming techniques that the peasants claim have greatly improved their crop yields.

I suppose the subordinating clause "the peasants claim" plays the role of an inserted sentence. If I take that out, the sentence will become:

The government is sending a team of agronomists to remote areas to look into cooperative farming techniques that have greatly improved their (the peasants') crop yields.

If my analysis is positive, I would like to ask whether the blue word "that" plays both the roles of...

1. the conjunction as well as the subject of have greatly improved their crop yields.
2. the conjunction as well as the object of the peasants claim.

Please advise.