This is a follow-up to this question on p-norms of tuples of operators.

Consider $\left[\begin{matrix} A \\ B \end{matrix}\right] \in B(H)^2_+$, meaning $A,B\geq 0$, and define $$ \left\|\left[\begin{matrix} A \\ B \end{matrix}\right]\right\|_p = \|(A)^p + (B)^p\|^{1/p}. $$ The previous question's answer conclusively showed that the generalization of the above is a norm only when $p=2$. However, the proof does not immediately transfer to this more limited situation (at least I can't see how to do it).

In this new case, it is obvious that this satisfies the triangle inequality on the positive cone for $p=1$ and $p=2$.

Q: For which $p$ does $\|\cdot\|_p$ satisfy the triangle inequality on $B(H)^2_+$?

If the answer is "no" for some $p$ then I would be very interested in a specific example as well as a more general result.

Edit: If this is established for any given p then one can create a norm via Haagerup's decomposition ideas.