Nginx - help with proxy configuration. Passing a request to the appropriate server

tl;dr How do I get https://prox1/client1/files to redirect to the appropriate client server?

Nginx on public facing server PROX1.
Three internal application servers: APP1, APP2, APP3. All running an in-house application, each dedicated to a specific client.
PROX1 has a share mounted locally, for each app server:

/mnt/shares/app1
/mnt/shares/app2...

The shares have the content we are serving to each client. What we would like is to have:
client1 use https://prox1/client1/files
client2 use https://prox1/client2/files and so forth.
MY QUESTION:
Will configuring the proxy_pass parameter in the location context accomplish this? Where the proxy_pass value has the full path to the respective server, or, since the files are available locally, do I not need to use that parameter?
OR
Should I try to do this all through the server context, forget about the NFS shares, and just create a server for each client?
pastebin config
Cheers