The strut-side perimeter conjecture for convex polygons

Let $n\geq 3$ and let $m\in\{1,\dots,n-1\}$. Let $P$ be a convex polygon with consecutive vertices $A_1,A_2,A_3,\dots,A_{n-1},A_n$. A side is said to have a strut when it satisfies the strut condition used in the preceding results. Suppose that the sides $A_1A_2,A_2A_3,\dots,A_mA_{m+1}$ have struts and that

$$|A_1A_2|+|A_2A_3|+\cdots+|A_mA_{m+1}|\geq 1.$$

Strut-side perimeter conjecture. The perimeter $L(P)$ satisfies

$$L(P)\geq 3.$$

The statement generalizes the preceding theorem for two adjacent sides and is motivated by examples whose perimeters approach $3$.