From the negative answer to this question we know that C$^*$-algebras that are isomorphic after tensoring with $M_n$ for all $n\geq 2$ need not be isomorphic. So what happens when we strengthen this?

In the following, if $\mathcal A\subset B(\mathcal H)$ and $\mathcal B\subset B(\mathcal K)$ then the minimum tensor product $\mathcal A\otimes_\textrm{min} \mathcal B$ is the closure of the algebraic tensor product $\mathcal A\odot \mathcal B$ in the $B(\mathcal H\otimes \mathcal K)$ norm.

Question 1:Let $\mathcal A,\mathcal B$ be C$^*$-algebras such that for all C$^*$-algebras $\mathcal C$ non-isomorphic to $\mathbb C$

$$\mathcal A\otimes_\textrm{min} \mathcal C \simeq \mathcal B\otimes_\textrm{min} \mathcal C $$ Does this imply $\mathcal A\simeq \mathcal B$?

Secondly, can one accomplish this with a single C$^*$-algebra.

Question 2:Does there exist a C$^*$-algebra $\mathcal C$ not isomorphic to $\mathbb C$ such that for all C$^*$-algebras $A,B$ we have $$ \mathcal A \otimes_\textrm{min} \mathcal C \simeq \mathcal B\otimes_\textrm{min} \mathcal C \ \Rightarrow \ \mathcal A \simeq\mathcal B\ ? $$

One can also ask these questions with the max tensor product.