Denote by $A$ the full ring of polynomials in one variable $t$ over the finite field with $q$ elements.
For any monic polynomial $P \in A$ define
$$
\sigma(P) = \sum_{d \mid P, d\, \text{monic}} d.
$$
A monic polynomial $P \in A$ is called `perfect`

if
$$
P = \sigma(P).
$$

Let $q=2.$
Two polynomials $P,Q \in A$ are called `consecutive`

if $\deg(P)>0$ and ($Q=P+1$ or $P=Q+1$).

In AMM problem 10771 $[1999,166]$ proposed by Florian Luca and resolved by Francis B. Coghlan it is proved the inexistence of consecutive perfect numbers.

Question: Let $q=2.$ Are there consecutive polynomials $P,Q \in A$ such that $P$ and $Q$ are both perfect.

EDIT. Changed incorrect $A$ by correct $P$ in the definition as observed by Valerio.