SOLVED

Let $1\leq a_1<\cdots<a_k\leq x$. How many of the partial products $a_1,a_1a_2,\ldots,a_1\cdots a_k$ can be squares? Is it true that, for any $\epsilon>0$, there can be more than $x^{1-\epsilon}$ squares?

Erdős and Graham write it is 'trivial' that there are $o(x)$ many such squares, although this is not quite trivial, using Siegel's theorem.

A positive answer follows from work of Bui, Pratt, and Zaharescu [BPZ24], as noted by Tao in this blog post. In particular Tao shows that, if $L(x)$ is the maximal number of such squares possible, and $u(x)=(\log x\log\log x)^{1/2}$, then \[x\exp(-(2^{1/2}+o(1))u(x)) \leq L(x) \leq x\exp(-(2^{-1/2}+o(1))u(x)).\]

See also [841].