Nonexistence conjecture for the degeneration $B^{(1)}_3\not\to C_{6,6}$

Consider the degeneration relation $\to$ among Jordan nets in $\operatorname{Gr}(3,\operatorname{Sym}^6)$, and the Jordan nets $B^{(1)}_3$ and $C_{6,6}$.

Nonexistence conjecture. For $n=6$, one has

$$B^{(1)}_3\not\to C_{6,6}.$$

The conjecture would imply that the degeneration diagram in Figure 1 of the source is complete. The source expresses this as a belief and gives no resolution, so it remains open.