Is Engineered wood okay on top of cement board?

We had our 1960s step down living room raised (about 3.5 inches) as part of a bigger renovation project, but when it was raised the builders screwed down cement board on top of plywood to finish it off. I'm not exactly sure why because there is no thin set under the cement board but I assume they were trying to have a similar material to our cement slab. Unfortunately the raised floor is anywhere from 1/2 inch to a 1/4 short in places and we are trying to even it out to match with the adjacent cement subfloor. Would it be okay to put engineered wood flooring over the cement backer board? We are considering putting thin-set under the cement board to see if that helps with the height or would it be better to take it off and put down a thicker piece of plywood on top of the other one?

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