The boundary-isoperimetric conjecture for triangular grids

Let $P_n$ be the triangular grid graph with $n$ vertices on each side, and let $\partial C$ denote the boundary vertices of a vertex set $C$. A row packing fills each row completely before filling the next row, from left to right; an ice cream cone packing fills diagonal rows from the lower-left corner in the order specified in the source. Boundary-isoperimetric conjecture. For every vertex set $C$ in $P_n$, $|\partial C|$ is at least the smaller of the number of boundary vertices in a row packing with $|C|$ vertices and the number in an ice cream cone packing with $|C|$ vertices. This is proposed as an isoperimetric inequality that would support the triangular-grid clearing conjecture.