Existence of a valid norm with sublinear type-2 growth

Let $X$ be a norm on the algebraic tensor product $\mathcal{S}_1^{\tilde k'\!, n}\otimes\mathcal{S}_1^{\tilde k'\!, n}\otimes\ell_2^{k\tilde k'}$. Call $X$ valid if

$$\|x\|_X\gtrsim\|x\|_{X^{iii}}$$

for every $x$, and

$$\left\|\lvert i\rangle\otimes\lvert j\rangle\otimes\left(V\lvert ij\rangle\otimes\operatorname{Id}_{\ell_2^{\tilde k'}}\right)\right\|_X\leq 1$$

for every $V\in B_{M_{n^2k,\tilde k'}}$ and $i,j=1,\ldots,n$. Valid-norm conjecture. There exists a valid norm $X$ and a dimension-independent constant $\beta<1$ such that

$$\mathrm{T}_2^{(n^2)}(X)\lesssim_{\log}n^\beta.$$

Any such norm would yield the type-2 estimate required by the preceding argument and hence the intended cryptographic lower bounds. The source presents this as an even weaker conjectural form and supplies no resolution.