Nope, it won't work the other way. I had heard both stories years ago, and thought the same as you. Then I heard the "7.7 in Springfield" story from a source who is almost always accurate and decided to try it. I admit to being a bit surprised at the result. Not only did the 7.7 (Norma) rounds load from the Japanese clip but it fired fine and I could not tell it from the .30-'06. I fired 20 shots and got a 100 yard group of about 6 inches. (I checked with the scope, I didn't pick up the target.) With my aging eyes and iron sights, and wrong ammo, I thought that was pretty good.
Two points of interest. The "oversize bullet" (.311 vs .308) did not blow up the rifle, cause the bolt to come back into my head, or wipe out the county. There was, in fact, no sign of any significant pressure increase; the primers looked quite normal.
The supposed case swelling didn't happen either, nor did it with reworked .30-'06 fired in a Japanese rifle. The base dimension for the .30-'06 is .470 +/- a thousandth or so. The base dimension of the 7.7 Japanese is .471 again +/-. In practical terms, they are the same. So blown cases, bulging brass, go the same route as other myths.
There is probably some confusion from the fact that the Japanese used three different 7.7mm cartridges.
1. The rimless 7.7x58, which we are discussing. It was used in the Type 99 rifle and the Type 92 light machine gun.
2. The 7.7 semi-rimmed was used in the Type 99 heavy MG, a Hotchkiss type; the feed strips are around at gun shows occasionally. Some folks buy them thinking to use the ammo in their rifles; won't work.
3. The thrid 7.7 round is rimmed; it is simply the .303 British made for use in the Japanese Navy's licensed copies of the Lewis gun.