Will I have problem with PHP 5.5 and WordPress 4.9.4?

No, you will not face any problem because of PHP 5.5. WordPress recommends (not required or mandatory) PHP 7.2 because it has better performance than the older versions. If you need better performance than PHP 5.5 then you should definitely use PHP 7.2. Btw, WordPress even smoothly works with PHP 5.2.4!.

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