Little-o upper-bound conjecture for equal-weight separating hash families

Let $C(N,q,\{w_1,\ldots,w_t\})$ be the maximal size of an $SHF(N;n,q,\{w_1,\ldots,w_t\})$. Let $t\ge 3$ be a positive integer, set $w_1=\cdots=w_t=1$, and assume $(t-1)\nmid N$.

Little-o upper-bound conjecture.

$$C(N,q,\{1,\ldots,1\})=o\left(q^{\left\lceil\frac{N}{t-1}\right\rceil}\right).$$

The conjecture concerns the nondivisible-length case, where the known general upper bound may or may not be of the correct order. The source notes that the claim is known when $N=t\ge 3$, while the general case remains open; attempted use of the hypergraph removal lemma did not resolve it.