Just wanted to provide an update on my experience with this issue. I spoke with a very helpful support specialist from Network Solutions this morning. They were able to confirm that their shared hosting environment servers do not allow outgoing PHP connections, which is why the connection to the update server failed. They were not able/willing to activate this feature in the shared hosting environment, but they said it would be a possible configuration for a virtual private server hosting package. Due to the increased cost of that kind of package, my client simply those to move their site to a host that could accommodate outgoing PHP connections.
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