The page doesn't render very well for me, so it's hard to tell what's going on. There's lots of "[Math Processing Error]" garbage.

Here's how I'd approach solving it. First, I'd split it into two subproblems, each sending two bits.

The first two bits are straightforward. B makes a Bell pair and sends half to A and half to Sender. A forwards its entangled qubit directly to the receiver. Sender superdense codes two bits and sends the resulting qubit to the receiver via C. The receiver decodes the two bits the usual way. This costs one qubit each on the B->A, A->Receiver, B->Sender, Sender->C, and C->Receiver links.

Now for the other two bits. B generates a Bell pair and sends half to C via A and half to Sender. Sender superdense codes the remaining two bits and sends the coded qubit to C. C decodes the two bits. Receiver generates a Bell pair, keeps half, and sends half to C via A. C superdense codes the two bits it just decoded into the entangled qubit it got from Receiver and sends the coded qubit to Receiver. Receiver decodes it. This sends one qubit each B->A, B->Sender, Sender->C, C->Receiver, and Receiver->A as well as two qubits A->C.

Adding it up, this uses exactly the allowed capacity on all links. Problem solved using standard techniques.

Edit: fixed a typo

I'm not even going to pretend I understood any of that