It is unlikely to "jeopardize;" however you will have to make some decisions regarding residency sooner than later.
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No. H-1B is a "dual intent" visa, i.e. both immigrant and non-immigrant in a sense, meaning that although H-1B is only a non-immigrant visa, you can still harbor immigrant intent at the same time. But ultimately the best avenue may be your marriage to a US citizen "if" this is a bona fide marriage despite being a commmuter marriage.