You can use the same approach as in the specified article and send a message to it.
Instead, use const int CB_SHOWDROPDOWN = 0x14F for your message.
From this reference sample, the bit was changed:
Message.Create(comboBox.Handle, CB_SHOWDROPDOWN , (IntPtr)1, IntPtr.Zero); // to open Message.Create(comboBox.Handle, CB_SHOWDROPDOWN , (IntPtr)0, IntPtr.Zero); // to close
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