Per Ptil Tekhelet, the shitta of the Ba'alei Tosefot (following Rashi) is that half of the strings in the tzitzit are to be made of techeilet, as opposed to 1/4 (rov haposkim) or 1/8 (Rambam). However, Sifre Bamidbar (Parshat Shelach, והיה לכם לציצית) Shammai and Hillel are cited with Shammai holding that three strings plus one of techeilet are necessary and Hillel holding that only three total are required (these positions are reversed in the Gemara, Menachot 39 or 40).
How do we reconcile the opinions of Shammai and the Ba'alei Tosefot, considering their apparent exclusivity? Is Shammai's shitta for techeilet a minimum, or an absolute requirement?