I don't think there's any doubt that Captain Matt Leach of the Indiana State Police was the source of many of the post Dillinger shooting allegations. Leach no doubt was highly disturbed he wasn't included in the Dillinger apprehension and it's clear he had no love for Sgt. Zarkovich and others of the East Chicago, Ind. police.
In calling at one point for a "Congressional investigation" into the Dillinger affair it was Leach who alleged that the East Chicago Police wanted Dillinger killed to keep him quiet on corruption. His finger pointing went to the direction of Zarkovich, O'Neil and others at the East Chicago, Ind. police. Leach was "leaking" stories all along on a variety of issues and allegations.
In August, 1934, Hoover had asked Purvis about some of these allegations and specifically the one about wanting Dillinger murdered. This portion of the memo shown and Purvis' answer zeroes in on that issue and only that issue. The rest of the memo contains other information not germane to this immediate subject.
Purvis' answer on the question of murdering Dillinger is pretty clear. In any of Matt Leach's statements and allegations, it's necessary to keep his probable motive for them in mind.