I can’t frame allegiance without a substantial reason for such. For example, would/could any communication with relatives or other persons in the ‘old’ country be construed as having ties of some kind or another with foreign interests? I know about this because once many years ago when I applied for a defense job the matter came up in a interview for security clearance. The interviewer after looking over my application asked me if I had relatives in Russia. I told him that I no doubt had relatives in Russia because my mother had been in contact with my deceased father’s brother who was in Russia. The interviewer slapped me in the rhetorical face by bluntly asking ‘Why should you be trusted to not talk about confidential information with your relatives in Russia?’ My response was a reference to my brother’s and my service records. Apparently this was a satisfactory response as I was hired. This brings the point that allegiance has to be established as firmly and validly as possible in some situations but ever so more important in the case for POTUSA. Our Founding Fathers were very wise in this respect.
Your question is a good one. I wish there were a definitive answer provided for either within the Constitution, or through established case law. I can only surmise that “allegiance” could be interpreted either narrowly or broadly depending upon the situation and/or upon who the individuals who are interpreting the issue may be. The times we are living in today are decidedly different than at our founding. Those who “decide” the rules today are NOT our Founding Fathers.