The congruent skein relation for reformulated colored HOMFLYPT invariants

Let $\mathcal{L}$ be an oriented link, let $p$ be prime, and let $(\mathcal{L}_+,\mathcal{L}_-,\mathcal{L}_0)$ be the links obtained at a distinguished crossing. Let $\check{\mathcal{Z}}_p(\mathcal{L};q,t)$ denote the reformulated colored HOMFLYPT invariant associated with the row partition $(p)$ on every component. Write $[p]=q^p-q^{-p}$ and $\{p\}=\frac{q^p-q^{-p}}{q-q^{-1}}$, and interpret $A\equiv B\pmod C$ as $\frac{A-B}{C}\in\mathbb{Z}[(q-q^{-1})^2,t^{\pm1}]$.

Congruent skein relation. If the crossing is a self-crossing of a knot, then

$$\check{\mathcal{Z}}_{p}(\mathcal{L}_+;q,t)-\check{\mathcal{Z}}_{p}(\mathcal{L}_-;q,t)\equiv(-1)^{p-1}\check{\mathcal{Z}}_{p}(\mathcal{L}_0;q,t)\pmod{\{p\}^2}.$$

If instead the crossing links two different components of $\mathcal{L}$, then

$$\check{\mathcal{Z}}_{p}(\mathcal{L}_+;q,t)-\check{\mathcal{Z}}_{p}(\mathcal{L}_-;q,t)\equiv(-1)^{p-1}p[p]^2\check{\mathcal{Z}}_{p}(\mathcal{L}_0;q,t)\pmod{\{p\}^2[p]^2}.$$

The source attributes this proposed relation to earlier work and does not provide a resolution of it.