Existence of complex Hadamard matrices at every even order

Prove that a complex Hadamard matrix exists for every even positive integer order.

Background

The paper defines a complex Hadamard matrix of order c as a matrix with entries drawn from {+1, -1, +i, -i} satisfying CC* = cI_c. Such matrices extend the real Hadamard framework by allowing complex phases.

The paper explicitly presents universal existence at even orders as a conjecture. Establishing this claim would characterize the orders for which the specified class of complex Hadamard matrices exists.