The equal-slit-length conjecture for doubly-intruded hexagons

Consider the scaling regime in which $x+y=y+t=z=N$, all fern lobe sizes are equal to $1$, both ferns have an even number of lobes, and their combined length is $m+n=M$. Let the regular hexagon have side length $M/2+N$, and suppose that, as the parameters tend to infinity, $M/(M/2+N)\to p$ with $0<p<2$, $x/(M/2+N)\to h$ with $0<h<1$, and $m/n\to r$. Equal-slit-length conjecture. For any given $0<p<2$ and any fixed $0<h<1$, the maximum occurs when $r_p=1$ $($i.e., when the slits have the same length$)$. This predicts that the number of lozenge tilings is maximized when the two intruding slits have equal lengths, although the problem remains open in the stated scaling regime.