So, the jose library needs a little work done to separate the ES (Ephemeral-Static) and SS (Static-Static) operations.
The JWA RFC 7518 only defines ES operations and actually states (https://tools.ietf.org/html/rfc7518#section-4.6):
A new ephemeral public key value MUST be generated for each key agreement operation.
The separate modes of operation were clarified by a different working group, COSE WG, in RFC 8152 (https://tools.ietf.org/html/rfc8152#section-12.4):
Ephemeral-Static (ES) DH: where the sender of the message creates
a one-time DH key and uses a static key for the recipient. The