In a recent paper, S. Khot and A. Naor show a natural generalization of the positive semidefinite Grothendieck's inequality. Grothendieck showed that there exists a constant $K > 0$ such that for every $n \times n$, symmetric semidefinite matrix $A=[a_{ij}]$, the following inequality holds:

$$\max_{x_1,\ldots,x_n} \sum_{ij} a_{ij}x_i^Tx_j \le K \max_{\epsilon_1,\ldots,\epsilon_n \in [-1,1]} \sum_{ij}a_{ij} \epsilon_i\epsilon_j,$$
where each $x_i$ is a *unit vector* (in Euclidean norm, so that $x_i^Tx_i=1$).

Khot and Naor studied a natural variant of this inequality, where the $n$ numbers $\epsilon_i$ are replaced by $n$ vectors $u_1,\ldots,u_n$ chosen from a set of $k < n$ unit vectors $v_1,\ldots,v_n$. The inequality becomes:

$$\max_{x_1,\ldots,x_n} \sum_{ij} a_{ij}x_i^Tx_j \le C \max_{u_1,\ldots,u_n \in \{v_1,\ldots,v_k\}} \sum_{ij}a_{ij} u_i^Tu_j.$$

They proved that this inequality actually holds, but the proof looks very complicated; as does the constant $C$. My question is thus in two parts:

(a) Is there any chance that there is a simpler proof for this "natural" generalization to Grothendieck's inequality?

(b) Does it seem like a feasible project to try and estimate this constant?

I am not an expert in these inequalities, but this generalization looks so interesting, that I had to seek out expert opinion.