By the time those boxes are ready to be recycled the facility to recycle will have arrived. Give the box 50 years at a minimum, but ask also
what will determine the need to recycle a poly box? If dropped it will break and can be glued; if eaten by rodent it can be filled; if painted, the UV will not affect it.
Fifty or so years is just the current known period of use, and after another fifty we may see that the need to recycle is unnecessary, that it certainly ought to be so, and will certainly not be as our current imagination dictates.
In this respect the poly hive will be seen as durable, a facet of human consumption which we have been trained to believe is anathema. After all, how would capitalism thrive without repeat consumers? I bought a toaster in 1985 for £40 and used it yesterday, and in that period fitted two sets of elements. During those 37 years others will have bought 9 nine toasters, throwing each away as in-built obsolesence arrived.
Durability ought to guide our future consumption, but while advertising is allowed to oil the wheels of vanity and thoughtless desire for the new, efforts to reduce environmental damage way into the future will be compromised.
Forty years ago I drove a Morris Minor, a miracle of durable engineering, and today a 1975 Land Rover Series with no heater. These tools were designed to do a job for a very long time, will not need to be recycled and are economical to maintain; I venture that the poly hive will be seen in a similar light, and last a lifetime.
Next time you put the kettle on (or need to buy another) have a read of this piece, written to celebrate architect Charles Ware, restorer of the Georgian house and Morris Minor, prophet of durability and 40 years ahead of his time.
https://driventowrite.com/2015/10/31/the-durable-car-morris-minor-charles-ware/