Anyway, I will try with an additional pin later today, but I doubt about it since it is failing at ACMD41 and previous transactions are succeeding.
I've added the CS pin to give it a try : as expected, it didn't help, especially that it been asserted to LOW at the begin of the card.init() and remain LOW until the time-out occurs... (So, equivalent to having it tied to GND)
Still hanging on ACMD41, but what strange is that during tons of captures I've done, sometime I've got and an answer R1_IDLE_STATE, but since we are waiting for R1_READY_STATE, the ACMD41 is sent again, then silence comes back as before ...
This silence is make me sad. I thought about potential power issue, I added caps, no success !
I've switched the adapters between my ESP and my STM32, no success !