## Trying to obtain a numeric solution to an inequality. Code might be not working as intended

I need to check whether an inequality can have solutions in the real numbers, given some extra constraints on the variables. I suspect that no solutions exist but I would like some verification of that claim. My code is the following:

ineq := {(2/9) * (l1 + l2 + l3)^2 > RankedMax[   Eigenvalues[    Transpose[{{t1 + l1, t2, t3}, {-t1, -t2 - l2, -t3}, {t1, t2,         t3 + l3}}].{{t1 + l1, t2, t3}, {-t1, -t2 - l2, -t3}, {t1,        t2, t3 + l3}}], 1]^2 +   RankedMax[   Eigenvalues[    Transpose[{{t1 + l1, t2, t3}, {-t1, -t2 - l2, -t3}, {t1, t2,         t3 + l3}}].{{t1 + l1, t2, t3}, {-t1, -t2 - l2, -t3}, {t1,        t2, t3 + l3}}], 2]^2}; con1 = {1 >= l1 >= -1}; con2 = {1 >= l2 >= - 1}; con3 = {1 >= l3 >= -1}; 

In practice I have four more constraints on the $$t_i$$ and $$\lambda_i$$ but to keep things simple I have omitted them. Now, since I am not optimistic that the Reduce command could track the problem, I use NSolve instead. It would be great if I can find even one solution to this inequality and not a general characterisation of the solutions, if they exist (I doubt it). Thus, I run the command:

NSolve[{ineq[[1]], con1[[1]], con2[[1]], con3[[1]]}, {l1, l2, l3, t1, t2,t3}] 

However the program runs forever and even if I simplify it by letting the $$t_i$$ be equal to zero, in which case I know there are no solutions to the inequality by a slightly different analysis, I still get no quick answer. I try the simple case with the following command:

NSolve[{ineq[[1]], con1[[1]], con2[[1]], con3[[1]], t1 == 0, t2 == 0,  t3 == 0}, {l1, l2, l3}] 

This makes me think that my code is not very good and I am wondering if there is something that can be done to make this actually work.

$$\|fg\|_{L^{p, q}} \lesssim_{p_1, p_2, q_1, q_2} \|f\|_{L^{p_1, q_1}}\|g\|_{L^{p_2, q_2}}$$
for all $$0 < p, q, p_1, q_1, p_2, q_2 \leq \infty$$ such that $$1/p = 1/p_1 + 1/p_2$$ and $$1/q = 1/q_1 + 1/q_2$$.
My question is whether anything is known about the best dependence on the exponents, and in particular best dependence on $$p_2$$ asymptotically for $$p_2$$ very large?