If you haven't seen the amazing community HubConnect app yet, it can do modes from HE to ST (or HE to HE), among many other things. It is, however, a server-to-client mode push. The other way does not work natively. I don't think the older, community Other Hub app ever added modes. The native Hub Link app (or, more specifically, the ST Send Hub Events app side of this) can send modes from ST to HE, but like all of Hub Link, that's unidirectional with ST.
However...not all hope is lost. What you can do is create a virtual switch for each mode, then use an automation to sync the modes and switches. Rule Machine can do this, but I think even Mode Manager can do this now too (turn switches off on mode change--the other switches--and change mode when a specific mode switch turns on). That would be on the Hubitat side. On the ST side, I think you'll have to use webCoRE or another custom app to sync the two. I had something like this going for a while if you care to know more, but if you're well-versed with both you can probably figure something out that works well for you.
If you can find a way to change mode on Hubitat only, it's a lot easier. 