Let $A_1$ and $A_2$ be $*$-algebras such that, for $i=1$ or $2$ and each $a\in A_i$,
$$||a||_{i,\text{max}}:=\sup_{\phi,H_\phi}\left\{||\phi(a)||_{H_\phi}:\phi\text{ is a $*$-homomorphism } A_i\rightarrow\mathcal{B}(H_{\phi})\right\}$$
is finite. In other words, $A_1$ and $A_2$ have well-defined maximal $C^*$-algebra completions, which I'll denote by $A_{1,\text{max}}$ and $A_{2,\text{max}}$.
Question: Is it true that the algebraic tensor product $A_1\odot A_2$ also has a well-defined maximal completion, and that for all $x\in A_1\odot A_2$, we have
$$||x||\leq ||x||_{A_{1,\text{max}}\otimes A_{2,\text{max}}},$$
where on the left we have the norm of $x$ in the maximal completion of $A_1\odot A_2$ and on the right we have the norm of $x$ in the (maximal) tensor product of $A_{1,\text{max}}$ and $A_{2,\text{max}}$?