I don't think the Allies will have the ability to begin a major attack in July. A tactical attack in a limited front as a show of support - e.g. seize a few hills and better secure a pass, then yes. Their immediate target will be the Dodecanese Islands- the RN will demand it as they threaten the supply lines. Their first priority in OTL planning was to be defensive in the mainland and attack the islands. If I may make a suggestion, I find it more plausible as a show of support to engage in a limited attack in the mainland while they seize the Dodecanese in summer 1941.
Maybe with an early Mincemeat style deception to convince the Axis that the main thrust is to be on the Mainland.