Could seven First Fleet horses sustain a healthy population?

Could the First Fleet's seven horses form a genetically healthy population over many generations?

No. The original herd was only a beginning, and many more horses had to be brought to Australia to avoid severe inbreeding.

The initial herd consisted of one stallion, one colt, three mares, and two fillies loaded at the Cape. That was enough to introduce horses to New South Wales, but it could not provide a genetically sustainable population for many generations. The colony had to import many more horses to avoid severe inbreeding. Establishing livestock permanently therefore required a much larger transport effort than the symbolic first arrivals suggest.