SMTP standards : needs outgoing SMTP server be MX for my domain?

[DEFFONTAINES Vincent <Vincent.DEFFONTAINES@coe.int>]

An organization refuses emails from my domain, under this reason :
My domain's mailer that connects to their SMTP server is not MX of my
domain.

Indeed it is not, I have different hosts for ingoing and outgoing email
traffic.
Actually, I see no good reason why outgoing mailer should be the same as MX.

I am wondering if SMTP standards require that email sender of a domain be
its MX? I find that really surprising.

I'll be glad to know if anyone can enlight me.

Thanks,

Vincent