Exactly. And the only reason June and farther contracts are priced higher was because of the expectation of a very short low price shock in spot. As there is a mathematical relationship between futures as well as forwards and spot those near month prices converge to spot at expiration. The pnl is purely realized within same month expires. The price difference between different expires does not factor at all into pnl unless someone trades the spread outright which is not the case here.
I get your point about "The futures curve remains constant", but it's the spot price which the front month price converges to at expiration (Which does not usually appear in the futures curve mostly).
Even if the futures curve remains constant in spreads between expires, you buy Jun CL for $35 & Jul CL is at $40, your Jun CL will settle somewhere down near WTI spot at say $15 (By that time maybe Jul CL will be at $20 (Same $5 difference), which can settle later at any price ($10 maybe, $20, $25, you guess).
Besides, the futures curve is never static, the curve is always moving & the spreads are very dynamic![]()