Forecast model developed now, but only used later, is it still valid?

I am not sure what the appropriate title for my question is, so I am open to suggestion/correction.

Now, the question:

• I observed a daily series $X_t, t=1,...,T$ also series $Y_t, t=1,...,T$.
• My aim is to use $X$ to predict $Y$ in the future. Specifically, I will observe $X_t, t=T+k,...,T+K$ where $k$ can be large. I will not observe any more $Y_t$. But I would like to estimate it.
• So I fitted a regression with ARIMA error using $(X,Y)_t,t=1,...,T$. (this is done using the forecast package.)
• The question is, (a) since new data won't come until some time later ($k$ being large), is any forecast still valid? (b) Even if it is valid, how do I forecast anyway? Do I start from $t=T+1$ and work my away towards $t=T+k$? This seems inefficient because I don't care about $t=T+1,...,T+k$. Furthermore, this isn't really possible because $X$ is not observed during that gap period.(I suppose one can argue you can forecast $X$ using some other model to fill the gap, but this still doesn't solve the problem that it is forecasting things I don't care about, let alone part (a) of my question)

So, what should I do?

P.S. Extra info. We can assume $T$ is large enough to cover any periods multiple times.