The stronger factor-sequence conjecture for rectangle diagrams

Let $\gamma$ be a path through a rank diagram, and let $T$ be the rectangular tableau associated to a triangle where $\gamma$ has an angle pointing down. Suppose the corresponding rectangle has $a$ rows and $b$ columns. For tableaux $X$ and $Y$ with entries strictly larger than those of $T$, with $Y$ having at most $a$ rows, write $S\left(\frac{T\mid Y}{X}\right)$ for the associated symmetric function. Let $(\dots,X,Y,\dots)$ be a factor sequence for $\gamma$, where $X$ and $Y$ label the displayed line segments. If $(X,Y)$ does not fit around $T$ and $S\left(\frac{T\mid Y}{X}\right)\ne0$, and if $X'$ and $Y'$ are obtained by applying Fomin's involution as described in the source, then $(\dots,X',Y',\dots)$ is also a factor sequence for $\gamma$.

The stronger factor-sequence conjecture. Under these hypotheses, $(\dots,X',Y',\dots)$ is also a factor sequence for $\gamma$.

This conjecture is presented as a stronger statement implying Conjecture 1A. It concerns the combinatorics of rectangle diagrams and factor sequences; the supplied text does not give evidence that it has been proved or disproved.