Switch invariance conjecture for the HOMFLY polynomial

Suppose that a planar diagram of an oriented link $L$ intersects the vertical line $x=0$ at $(0,-b)$, $(0,-a)$, $(0,a)$, and $(0,b)$, where $0<a<b$. Suppose that $L$ is directed to the right at $(0,\pm b)$ and to the left at $(0,\pm a)$. Define the switch $L'$ by taking the $x>0$ part of $L$, reflecting it across the $x$-axis, and flipping all crossings. Switch invariance conjecture.

$$P(L;a,z)=P(L';a,z).$$

The conjecture proposes that this local switch operation preserves the HOMFLY polynomial, and is motivated by the plabic-graph invariance conjecture. The supplied source gives no resolution status.