The top-degree dimension conjecture for the exterior graded Swiss-Cheese operad

Let $V$ be a finite-dimensional vector space over $k$, and write $d=\dim_k(V)$. Let ${\Lambda}^{S^2}_V(m)$ denote the degree-$m$ component of the exterior graded Swiss-Cheese operad associated with $V$. Top-degree dimension conjecture. If $\dim_k(V)=d$, then

$$\dim_k\bigl({\Lambda}^{S^2}_V(2d+1)\bigr)=1.$$

The preceding results establish that ${\Lambda}^{S^2}_V(m)=0$ for $m>2d+1$ and prove the conjectured dimension in the case $d=2$; the assertion for arbitrary finite dimension remains open.