Kanade–Russell conjecture modulo 16 for restricted partition sets

Let $F$ be the set of the following 13 partitions:

$$\begin{gathered} (1,1,1),(2,1,1),(2,2,1),(3,2,1),(3,3,1),(5,3,3),(4,4,1,1),\\ (5,4,1,1),(5,4,2,1),(5,5,2,1),(6,5,3,1,1),(6,6,3,1,1),(7,6,4,2,1). \end{gathered}$$

Define

$$I_1=\{(1),(5,4,2,2),(9,8,6,4,2,2)\},\quad I_2=\{(1,1),(2,2),(4,3,1)\},$$ $$I_3=\{(1,1),(2,1),(2,2),(3,2),(3,3),(4,3,1),(4,4,1),(5,4,2),(6,5,3,1)\}.$$

For $1\leq a\leq3$, let $\mathsf L_a$ be the set of partitions that do not begin with any $\boldsymbol c\in I_a$ and do not match $(b_1+k,\ldots,b_p+k)$ for any $(b_1,\ldots,b_p)\in F$ and $k\geq0$. Kanade–Russell conjecture modulo 16. The partition sets satisfy

$$\mathsf L_1\stackrel{\mathsf{PT}}{\sim}T^{(16)}_{2,3,4,5,11,12,13,14},\qquad \mathsf L_2\stackrel{\mathsf{PT}}{\sim}T^{(2)}_1,$$ $$\mathsf L_3\stackrel{\mathsf{PT}}{\sim}T^{(16)}_{1,4,6,7,9,10,12,15}.$$

The restrictions exclude specified initial subpartitions and translated patterns, while the right-hand sides impose congruence conditions on all parts. The source presents these identities as a conjectural example connected with vertex-operator bases; their resolution status is not specified here.