If $A$ is a Banach agebra and $M$ is a Banach $A$-bimodule then a linear map $T:A\to M$ is called an $A$-module homomorphism if $$T(ab)=aT(b),\quad T(ab)=T(a)b,\qquad a,b\in A.$$ Also $A\hat{\otimes} A$ is a two sided $A$-module with the following actions
$$c(a\otimes b)=ca\otimes b,\quad (a\otimes b)c=a\otimes bc.$$
**(Here $\hat{\otimes}$ deotes the projective tensor product.)**

Does there exist a Banach algebra $A$ and a net of linear $A$-module homomorphisms $\{\rho_\alpha:A\to A\hat{\otimes} A\}_{\alpha\in I}$ such that for the linear product map $\pi:A\hat{\otimes} A\to A,$ $\pi(a\otimes b)=ab$ we have:

$\pi\circ\rho_\alpha(a)\to a$ in norm for all $a\in A$;

the set $\left\{\frac{\|\pi\circ\rho_\alpha(a)b-ab\|}{\|ab\|}:a,b\in A,\alpha\in I\right\}$ is bounded;

the set $\left\{\frac{\|\pi\circ\rho_\alpha(a)-a\|}{\|a\|}:a\in A,\alpha\in I\right\}$ is not bounded.