  The acyclic edge colouring problem is extensively studied in graph theory.
The corner-stone of this field is a conjecture of Alon et.
al.\cite{alonacyclic} that $a'(G)\le \Delta(G)+2$. In that and subsequent work,
$a'(G)$ is typically bounded in terms of $\Delta(G)$. Motivated by this we
introduce a term $gap(G)$ defined as $gap(G)=a'(G)-\Delta(G)$. Alon's
conjecture can be rephrased as $gap(G)\le2$ for all graphs $G$. In
\cite{manusccartprod} it was shown that $a'(G\Box H)\le a'(G)+a'(H)$, under
some assumptions. Based on Alon's conjecture, we conjecture that $a'(G\Box
H)\le a'(G)+\Delta(H)$ under the same assumptions, resulting in a
strengthening. The results of \cite{alonacyclic} validate our conjecture for
the class of graphs it considers. We prove our conjecture for a significant
subclass of sub-cubic graphs and state some generic conditions under which our
conjecture can be proved. We suggest how our technique can be potentially
applied by future researchers to expand the class of graphs for which our
conjecture holds. Our results improve the understanding of the relationship
between Cartesian Product and acyclic chromatic index.

Combinatorics;Discrete Mathematics