I suspect you’re right. From my hometown’s local waste authority: After [we have] collected the waste from your bin, the purple dotted bag [with plastic waste] is sent on for collection and pressing before being transported to large sorting facilities in Northern Germany and other places in Europe. Here, 75% of the content in the purple dotted bag (figures from 2019) is recycled, and the remaining waste (often residual waste and food waste that has been incorrectly sorted) is utilized for energy.

And more detail: This is how household plastic is recycled: Collected plastic packaging is first finely sorted in Germany, before moving on to recyclers in the same country or elsewhere in Europe. We differentiate between the recycling of household plastic, which is the plastic you sort at home, and plastic packaging from businesses. - The plastic packaging you sort at home is collected by the municipality or an inter-municipal waste company. They press the plastic together into large bales. - Plastretur transports the plastic to a sorting facility. - At the sorting facility, incorrect sorting and contaminants are removed. This can include paper and metal, or other items that ended up in the wrong collection at your home. - Additionally, the plastic is sorted into different qualities, and non-recyclable packaging is removed. The sorting facilities classify the plastic into several plastic qualities such as PP, LDPE, HDPE, PET before it is transported to a recycling plant. - At the recycling plant, the plastic is washed. This process also removes labels and glue. - Then it is ground and melted into small beads, called pellets. Pellets are a raw material that can be used to make new products from recycled plastic.