Strongest type-2 constant conjecture for operator-space tensor products

Let $n$, $k$, and $\tilde k'$ be the parameters in the construction, let $\mathcal{S}_1^{\tilde k'\!, n}$ denote the corresponding trace-class space, let $\otimes_{(\varepsilon,\pi)_{1/2}}$ and $\otimes_\varepsilon$ be the tensor norms used in the paper, and let $\ell_2^{k\tilde k'}$ be Euclidean space. For a Banach space $X$, write $\mathrm{T}_2^{(m)}(X)$ for its type-2 constant restricted to $m$ vectors, and write $\lesssim_{\log}$ for the paper's logarithmic comparison. Strongest-form conjecture.

$$\mathrm{T}_2^{(n^2)}\left(\left(\mathcal{S}_1^{\tilde k'\!, n}\otimes_{(\varepsilon,\pi)_{1/2}}\mathcal{S}_1^{\tilde k'\!, n}\right)\otimes_\varepsilon\ell_2^{k\tilde k'}\right)\lesssim_{\log}\mathrm{T}_2^{(n^2)}\left(\mathcal{S}_1^{\tilde k'\!, n}\otimes_{(\varepsilon,\pi)_{1/2}}\mathcal{S}_1^{\tilde k'\!, n}\right)\lesssim_{\log}n^{3/4}.$$

This estimate would provide the type-2 bound needed for the paper's unconditional exponential lower bounds on resources in position-based cryptography attacks. The source presents it as a conjectural strengthening; no resolution is supplied.