。
She'll likely be staying at the Royal Hotel this Tuesday.
Literal
She [topic-は] this-coming Tuesday [on-に] [contrast-は] Royal-Hotel [at-に] is-staying [probably-でしょう].
~でしょう is the polite probability marker — 'probably, presumably, will likely.' It softens an assertion into a confident guess. Combined with the resulting-state ~ている, the prediction lands on a state ('will be in a stayed condition') rather than an event ('will stay'). The contrastive は in 火曜日には quietly hints at a comparison: maybe she's elsewhere on other days. ロイヤルホテル (literally 'Royal Hotel') is a generic name used by many properties across Japan and abroad — the prestige connotation of 'royal' has been borrowed wholesale into the hospitality industry.