(Definition: Facet = Maximal Face)

This question is a continuation of the previous one that I had asked a couple of years ago: Is Euler characteristic of a simplicial complex upper bounded by a polynomial in the number of its facets ?

As an answer to my previous question, David Speyer gave an example showing that the Euler characteristic of a simplicial complex need not be polynomially bounded in terms of number of its facets. He further suggested a weaker bound, which if true would come handy in my research. Hence, I am posting it as a question here.