Conjecture for the dimension function $\mathfrak{f}(s,t)$

Let $\mathfrak{f}(s,t)$ denote the dimension quantity defined in the paper, with $s$ and $t$ in the parameter ranges below. The conjecture is the dimension-function conjecture: for $s\in(0,\tfrac{1}{2}]$ and $t\in[s,3s]$, or for $s\in[\tfrac{1}{2},1]$ and $t\in[s,2-s]$,

$$\mathfrak{f}(s,t)=\frac{3s+t}{2}.$$

This is proposed as a plausible numerological analogue of the Furstenberg set problem. The precise value of $\mathfrak{f}(s,t)$ is stated to be unknown in the remaining cases, so the conjecture remains open.