I assumed constant fuel flow which wouldn't happen on a carbureted engine when the OAT changes. It may happen however on injected engines where there are corrective measures for varying density but we all know how good it works from time to time...
E.g. You are operating peak egt. You increase MAP with const FF. Would the power change? No. There is simply no fuel left to be burned. Same figure for decreasing temperature. More mass with at constant MAP put into the engine thus operating on the lean side. Now if the fuel controller manages to compensate for the change in density altitude (changing temperature on a constant pressure alt) the power would increase.
I wouldn't be too sure about decreasing TAS, I'm not sure whether increased drag would weigh out increased power (condition: not operating on constant percentage of T/O power as in the POH).
My assumption for the given conditions on a fixed pitch prop would be increasing rpm, decreasing MAP as the power increases and the lower deck pressure drops with increasing RPM and volumeflow.