Suppose we construct sets $A,B,C$ with
$A \subseteq [0,1]$, $\dim_\mathrm{H}(A) = 1/5$ and $\dim_\mathrm{P}(A) = 4/5$,
$B \subseteq [2,3]$, $\dim_\mathrm{H}(A) = 1/5$ and $\dim_\mathrm{P}(A) = 2/5$,
$C \subseteq [4,5]$, $\dim_\mathrm{H}(A) = 1/5$ and $\dim_\mathrm{P}(A) = 3/5$.
Let $E = A \cup B, F = B \cup C$, so that $E \cap F = B$.
Then
$\dim_\mathrm{H}(E) = \dim_\mathrm{H}(F) = \dim_\mathrm{H}(E\cap F) = 1/5$,
and
$\dim_\mathrm{H}(E)=4/5, \dim_\mathrm{H}(F)=3/5, \dim_\mathrm{H}(E\cap F) = 2/5$.