Sublogarithmic bound for partially shattering permutations

Let $f_k(n,t)$ be the extremal function governing partially shattering permutations, with integers $k$ and $t$ as parameters.

Sublogarithmic-bound conjecture. If

$$k\geq 3\qquad\text{and}\qquad t\leq 2^{k-1},$$

then

$$f_k(n,t)=o(\log n).$$

This is presented after the paper's discussion of the unknown asymptotic range of $f_k(n,t)$ and would rule out logarithmic growth in the indicated parameter range.