I have a $\otimes$-triangulated category $\mathcal T$ and two triangles in $\mathcal T$: $$ x_0\to x_1\to c_x\to \Sigma x_0\ \ \ \text{and}\ \ \ y_0\to y_1\to c_y\to \Sigma y_0. $$ Consider the following diagram, where the object $P$ is constructed as a homotopy push-out of the obvious maps $\Sigma^{-1}c_x\otimes y_0 \leftarrow\Sigma^{-1}c_x\otimes\Sigma^{-1}c_y\to x_0\otimes \Sigma^{-1} c_y$ (one can do this either as in Neeman's book, or assuming that there is some enhancement for $\mathcal T$): Is it true that the cone of the map $\varphi$ obtained this way is isomorphic to $x_1\otimes y_1$?

The question is relatively natural, so I hope that somebody with a little more working experience with $\otimes$-triangulated categories will be able to give a simple proof or a known counterexample in some concrete situation. If this is not known in general, I am happy to assume that $\mathcal T$ has some kind of enhancement (e.g., $\mathcal T$ is the homotopy category of a stable $(\infty,1)$-category, or $\mathcal T$ is the base of a strong and stable derivator), or even, to start with, to assume that $\mathcal T$ is the derived category of a commutative ring.

total cofiberthat can be computed as in the bonus part of my answer here (technically, that's the total fiber instead, but you just need to dualize the arguments). I suspect that the statement is also true in a generic ⊗-triangulated category, using the octahedral axiom in creative ways. $\endgroup$ – Denis Nardin Aug 2 '19 at 20:50