Sum of squared nearest-neighbor distances between points on the sides of a rectangle

Question

For positive real numbers $a,b$, let $R$ denote the $a\times b$ rectangle $[0,a]\times[0,b]$. Let $A_1,\dots,A_4$ be points on the sides of $R$, one point on each side. For each $j=1,\dots,4$, let $s_j:=\min\{\|A_j-A_k\|^2\colon k\ne j\}$.

Is it then always true that $$s_1+\dots+s_4\le2a^2+2b^2?$$

Here one may also assume that each of the four $\frac a2\times \frac b2$ rectangles $R_1:=[0,\frac a2]\times[0,\frac b2],R_2:=[\frac a2,a]\times[0,\frac b2], R_3:=[0,\frac a2]\times[\frac b2,b],R_4:=[\frac a2,a]\times[\frac b2,b]$ contains exactly one of the points $A_1,\dots,A_4$.

This question is what mainly remains to complete the answer to that related question.