The pyramid path-search equality for triangular query budgets

Let $Py(n)$ denote the pyramid graph of height $n$. For a positive integer $k$, let $pa_k(Py(n))$ and $si_k(Py(n))$ denote, respectively, the minimum number of rounds needed for path-search and sink-identification when at most $k$ queries can be asked in each round. For a positive integer $l$, set

$$s_l=1+2+\ldots+l.$$

Pyramid path-search conjecture. If $s_l=1+2+\ldots+l$ for some $l$, then

$$si_{s_l}(Py(n))=pa_{s_l}(Py(n))=\lceil n/l\rceil.$$

This extends the fully adaptive equality $si_1(Py(n))=pa_1(Py(n))=n$ to larger query budgets, asserting that both search problems have the same optimal round complexity when the budget is triangular. The supplied text does not indicate whether the assertion has been proved or disproved.