Does Haplotypecaller assign genotype for each allele separately?

Hi,
I read how haplotypecaller assign genotype in the following link, it says that for each position haplotypecaller will calculate the best genotype, so I wonder, if the positions are close, then will haplotypecaller calculate genotypes separately for them? if so, will the selected haplotypes for these two positions be different?
https://software.broadinstitute.org/gatk/documentation/article.php?id=4442