Signalling models and out of equilibrium beliefs
In my interpretation of the model, competitive firms imply that the wage always equals the expected productivity, which depends on the beliefs. Clearly, in any pooling equilibrium the on-path beliefs are equal to the prior such that the wage is simply $E[\theta]$. If, in your example, a worker sends the off-path messags $e=1$, the wage must be $w(e=1)=w_1=b_1 \theta_H + (1-b_1) \theta_L$, where $b_1$ is the belief in the information set of $(e=1)$ that the type is high. Since $b_1 \in [0,1]$, it must be that $w_1 \in [\theta_L,\theta_H]$.
This information set is not reached on equilibrium path such that we are free to determine the out-of-equilibrium belief $b_{1}$. Only $b_{0}=p_H$ is a PBE-consistency requirement.
The set of all no-education pooling PBE is then given by the following: 
$e_H=e_L=0$, 
$w_{0}=E[\theta]$, 
$w_{1}=b_{1}\theta_H + (1-b_{1})\theta_L$, 
$ b_{0}=p_H$, 
$b_{1}: b_{1}\theta_H + (1-b_{1})\theta_L -c_H\leq E[\theta].$
Note that the beliefs are part of the equilibrium. To get rid of equilibrium multiplicity, have a look at the numerous refinements such as the intuitive criterion that kills unreasonable out-of-equilibrium beliefs (and thereby unreasonable equilibria). The restriction on belief $b_1$ originates from the incentive constraint of the high type. We want to make sure that $\theta_H$ does not want to deviate get eductation, and because the wages are competitive they are implied by the off-path belief which we can set freely.
The idea to set $w_1&lt;\theta_L$ would violate the competitive nature of firms.
This sentence can be interpreted in two ways.

Wages are just exogenously set this way.
There are two firms (having the same beliefs) who both set a wage menu $(w_1,w_2)$ and a la Bertrand the worker always goes to the firm setting the higher wage.
For some off-path beliefs, one firm would deviate from $w_1&lt;\theta_L$ and may attract workers by setting a wage $w_1=b_{1}\theta_H + (1-b_{1})\theta_L - \varepsilon$ and thereby make a profit. Because you did not specify the off-path belief, we have no idea whether this deviation on the wage would be profitable.