Never mind, consulting the SYSLINUX documentation confirmed for
another time, native EFI boot is not supported even with secure boot
disabled. As I said already, Legacy Boot on HP Probook 4540s (my
latest purchase), does not work.

An alternative would be to use a minimal distribution to manage grub*.
Something like Puppy comes to mind, or fsmithred's distro. Currently,
I have Debian Jessie managing grub*; yes, unfortunately, it is with
systemd. During installation, I noticed it is a far cry from being a
minimal system as many packages I don't need were installed. I used
debootstrap for the installation and only root exists as a user.

Edward

--
If you can't explain it simply, you don't understand it well enough.

Albert Einstein