Yes, you can do a 1031 exchange to buy a property of lesser price.
Suppose you sell the property and find the replacement property of lesser value. All you need to do is to involve the qualified intermediary in the process to handle the proceeds and do the constructions work. In this case, the investor lends the money and gives it to the qualified intermediary for the construction work. The QI completes the construction work within 180 days.
Here, the qualified intermediary acts as an Exchange Accommodation Titleholder (EAT).
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