For $n\geq 5$, a pentagonal packing of size $t$ is a set of $t$ edge-disjoint pentagons (cycles of length five) in the complete graph $K_n$. A pentagonal packing $\Cal P$ is maximal, denoted as $MPP(n)$, if the complement of the union of all pentagons from $\Cal P$ is pentagon-free. The spectrum $S^(5)(n)$ for maximal pentagonal packings is the set of all possible sizes of $MPP(n)$. We formulate a conjecture on the structure of the spectrum $S^(5)(n)$, and prove the conjecture for all $n=40k+3$, $% k\geq 2$.